US7751491B2 - Code design method for repeat-zigzag Hadamard codes - Google Patents
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- H—ELECTRICITY
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- H03M—CODING; DECODING; CODE CONVERSION IN GENERAL
- H03M13/00—Coding, decoding or code conversion, for error detection or error correction; Coding theory basic assumptions; Coding bounds; Error probability evaluation methods; Channel models; Simulation or testing of codes
- H03M13/29—Coding, decoding or code conversion, for error detection or error correction; Coding theory basic assumptions; Coding bounds; Error probability evaluation methods; Channel models; Simulation or testing of codes combining two or more codes or code structures, e.g. product codes, generalised product codes, concatenated codes, inner and outer codes
- H03M13/2957—Turbo codes and decoding
- H03M13/296—Particular turbo code structure
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- H03M—CODING; DECODING; CODE CONVERSION IN GENERAL
- H03M13/00—Coding, decoding or code conversion, for error detection or error correction; Coding theory basic assumptions; Coding bounds; Error probability evaluation methods; Channel models; Simulation or testing of codes
- H03M13/03—Error detection or forward error correction by redundancy in data representation, i.e. code words containing more digits than the source words
- H03M13/033—Theoretical methods to calculate these checking codes
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- H03M—CODING; DECODING; CODE CONVERSION IN GENERAL
- H03M13/00—Coding, decoding or code conversion, for error detection or error correction; Coding theory basic assumptions; Coding bounds; Error probability evaluation methods; Channel models; Simulation or testing of codes
- H03M13/03—Error detection or forward error correction by redundancy in data representation, i.e. code words containing more digits than the source words
- H03M13/05—Error detection or forward error correction by redundancy in data representation, i.e. code words containing more digits than the source words using block codes, i.e. a predetermined number of check bits joined to a predetermined number of information bits
- H03M13/11—Error detection or forward error correction by redundancy in data representation, i.e. code words containing more digits than the source words using block codes, i.e. a predetermined number of check bits joined to a predetermined number of information bits using multiple parity bits
- H03M13/1102—Codes on graphs and decoding on graphs, e.g. low-density parity check [LDPC] codes
- H03M13/1191—Codes on graphs other than LDPC codes
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- H—ELECTRICITY
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- H03M—CODING; DECODING; CODE CONVERSION IN GENERAL
- H03M13/00—Coding, decoding or code conversion, for error detection or error correction; Coding theory basic assumptions; Coding bounds; Error probability evaluation methods; Channel models; Simulation or testing of codes
- H03M13/03—Error detection or forward error correction by redundancy in data representation, i.e. code words containing more digits than the source words
- H03M13/05—Error detection or forward error correction by redundancy in data representation, i.e. code words containing more digits than the source words using block codes, i.e. a predetermined number of check bits joined to a predetermined number of information bits
- H03M13/11—Error detection or forward error correction by redundancy in data representation, i.e. code words containing more digits than the source words using block codes, i.e. a predetermined number of check bits joined to a predetermined number of information bits using multiple parity bits
- H03M13/1102—Codes on graphs and decoding on graphs, e.g. low-density parity check [LDPC] codes
- H03M13/1191—Codes on graphs other than LDPC codes
- H03M13/1194—Repeat-accumulate [RA] codes
- H03M13/1197—Irregular repeat-accumulate [IRA] codes
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- H—ELECTRICITY
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- H03M13/00—Coding, decoding or code conversion, for error detection or error correction; Coding theory basic assumptions; Coding bounds; Error probability evaluation methods; Channel models; Simulation or testing of codes
- H03M13/35—Unequal or adaptive error protection, e.g. by providing a different level of protection according to significance of source information or by adapting the coding according to the change of transmission channel characteristics
- H03M13/356—Unequal error protection [UEP]
Definitions
- the present invention relates generally to coding techniques for communications and, more particularly, to code designing for repeat-zigzag Hadamard codes.
- low-rate coding schemes play a critical role.
- Traditional low-rate channel coding schemes include Hadamard codes and super-orthogonal convolutional codes. These codes offer low coding gain and hence their performance is far away from the ultimate Shannon limit.
- the ultimate Shannon capacity of an Additive White Gaussian Noise (AWGN) channel in terms of SNR per information bit is about ⁇ 1.6 dB for codes with rates approaching zero.
- AWGN Additive White Gaussian Noise
- low-density parity-check (LDPC) codes and the repeat-accumulate (RA) codes offer capacity-approaching capability for various code rates when the ensemble profiles are optimized. However, in the low-rate region, they both suffer from significant performance loss and extremely slow convergence with iterative decoding.
- BER bit-error-rate
- PCZH parallel-concatenated zigzag-Hadamard
- RZH repeat-zigzag-Hadamard
- an RZH code can be represented by its Tanner graph, based on which iterative decoding algorithms can be derived.
- two decoding schemes namely the turbo-like serial decoding and the LDPC-like parallel decoding
- IRZH irregular RZH
- the exact density evolution method for LDPC code design can be employed for the IRZH code optimization.
- the complexity is extremely high, even for the moderate rate LDPC codes. Density evolution with Gaussian approximation does not work for the design of low-rate RZH codes. Accordingly, there is a need for a design method for low-rate RZH codes.
- the signal-to-noise ratio threshold means that for any signal-to-noise ratio greater than this threshold, the BER of the code will go to zero or a very small value when the code length is enough long.
- a method for selecting a signal to noise ratio for a communications code includes obtaining extrinsic information transfer (EXIT) information for a repeat-zigzag Hadamard (RZH) code responsive to a Hadamard order and a signal to noise ratio, determining code parameters for an irregular repeat zigzag Hadamard (IRZH) code for a corresponding code rate in response to the obtained EXIT values, and repeating the step of obtaining the EXIT information for a different signal to noise ratio if the corresponding code rate is other than a selected rate.
- EXIT extrinsic information transfer
- RZH repeat-zigzag Hadamard
- IRZH irregular repeat zigzag Hadamard
- the step of obtaining EXIT information includes one of obtaining an EXIT curve for repeat-zigzag Hadamard code by Monte Carlo simulation using serial decoding or obtaining an EXIT function for parallel decoding of the repeat-zigzag Hadamard code by using equations.
- a method for selecting a signal to noise ratio for a communications code includes choosing a signal to noise ratio, selecting a Hadamard code order, obtaining extrinsic information transfer (EXIT) information for one of a serial decoding and a parallel decoding of a irregular repeat-zigzag Hadamard (IRZH) code responsive to the Hadamard order and the signal to noise ratio, determining a code ensemble for the irregular repeat zigzag Hadamard (IRZH) code for a related bit error rate in response to the obtained EXIT information, and changing the signal to noise ratio for repeated steps of obtaining the EXIT information and determining a code ensemble for a different signal to noise ratio if the related bit error rate is other than a target rate.
- EXIT extrinsic information transfer
- the EXIT information is a curve for the irregular repeat-zigzag Hadamard (IRZH) code obtained by Monte Carlo simulation using the serial decoding and the EXIT information is a function for parallel decoding the irregular repeat-zigzag Hadamard (IRZH) code using equations.
- IRZH irregular repeat-zigzag Hadamard
- FIG. 1 is a block diagram of a systematical repeat-zig-zag-Hadamard RZH encoder
- FIG. 2 is a block diagram unpunctured, zig-zag-Hadamard RZH encoder
- FIG. 3 is a tanner graph of an unpunctured (n,k) RZH code
- FIG. 4 is a block diagram of turbo-like decoding of systematic RZH codes
- FIG. 4A is a flow chart of the method for code design for serial decoding in accordance with the invention.
- FIG. 5 is a block diagram of parallel decoding of systematic RZH codes
- FIG. 5A is a flow chart of the method for code design for parallel decoding in accordance with the invention.
- FIG. 6 is a graph of an extrinsic information transfer function (EXIT) of information bits in BIAWGN channel;
- FIG. 8 is a graph of bite error rates BERs for un-punctured RZG codes, with an information block length of 65536 and a maximum iteration number of 150, in the case of serial decoding in accordance with the invention
- FIG. 10 is a graph of bite error rates BERs for un-punctured IRZH codes, with an information block length of 65536 and a maximum iteration number of 150, in the case of serial decoding in accordance with the invention
- FIG. 11 is a graph of BER performance of irregular RZH codes over BIAWGN channel, in the case of parallel decoding in accordance with the invention.
- FIG. 12 is a graph of BER performance of irregular RZH codes over BIAWGN channel, in the case of parallel decoding in accordance with the invention.
- the inventive code design considers two decoding schemes, namely the turbo-like serial decoding and the LDPC-like parallel decoding. Parameters of an RZH code need to be chosen carefully for capacity-approaching performance, which leads to the irregular RZH (IRZH) code with optimized degree profile.
- the invention considers the design methods for IRZH codes.
- the extrinsic information transfer function (EXIT) chart technique can be employed to optimize the code profiles.
- EXIT functions characterize mutual information between the input and output of constituent decoders significantly facilitates performance analysis of iterative coding schemes.
- the design method for both decoding schemes is based on EXIT function/EXIT chart. One may always get the EXIT by Monte Carlo simulation for both decoding schemes.
- EXIT function analytically, i.e., a mathematical formulation. With that one can calculate the EXIT value without using the simulation, which simplifies or reduces the complexity of the code design procedure significantly. Since in this case the inner decoder is a trellis-based zigzag-Hadamard (ZH) decoder, an analytical expression of its EXIT function is not available, and the inventive code design resorts to Monte Carlo simulations to obtain the EXIT functions of the inner code.
- ZH trellis-based zigzag-Hadamard
- connection bits of the ZH code are treated as degree-2 variable nodes, and the inner constituent decoder becomes a Hadamard decoder, having the channel observations of the parity bits and the a priori probabilities about the information or connection bits as the input, and the extrinsic information for the information/connection bits as the output.
- EXIT functions with multiple inputs and outputs are difficult to obtain using Monte Carlo simulations.
- EXIT functions of Hadamard with multiple inputs and outputs in the context of low-rate RZH codes over BIAWGN channels are derived and applied to the analysis and design of RZH codes. With the EXIT functions, bit error rates (BERs) for a given communication code profile can be estimated and the differential evolution (DE) method can be employed to optimize the degree profile.
- BERs bit error rates
- DE differential evolution
- the structure of a systematic, repeat-zigzag-Hadamard RZH code is shown by the diagram 10 in FIG. 1 .
- the RZH code is an alternative code structure to parallel concatenated ZH codes and turbo-Hadamard codes.
- R c r d v ⁇ ( 2 r - r - 1 ) + r , ( 1 ) where r is the order of the Hadamard code for the inner encoder.
- the un-punctured RZH codes output on a single channel 22 whose structure 20 is shown in FIG. 2 .
- a RZH code can be represented by its Tanner graph 30 , as shown in FIG. 3 , there are three sets of nodes in the Tanner graph of a RZH code: the variable nodes of information bits, corresponding to u, the variable nodes of common bits, corresponding to q, and the Hadamard check nodes, corresponding to the Hadamard code constraints. Note that in FIG. 3 , the parity bits of Hadamard code words are embedded in their Hadamard check nodes. Those information bits that are repeated d v times are represented by information bit variable nodes with degree d v , since they participate in d v Hadamard code constraints, and the common bit variable notes can be viewed as degree-2 information bit variable nodes.
- Each Hadamard check node is connected to r information bit nodes and to two common bit nodes.
- the connections between Hadamard check nodes and information bit nodes are determined by the edge interleaver and are highly randomized, whereas the connections between Hadamard check nodes and common bit nodes are arranged in a highly-structured zigzag pattern.
- BP belief-propagation
- Both decoding schemes entail outer repetition decoders 44 and inner serial decoder 41 , 51 or a Hadamard decoder 51 with interleavers ⁇ ⁇ 1 42 and ⁇ 43 .
- Output from the repetition decoders is sent to a decision block 45 , tied to the sink output 46 .
- the serial decoding takes less iteration number than the parallel one, whereas the latter enjoys the advantage of easy hardware implementation with parallel processing. There is no fundamental difference in terms of bit error rate (BER) performance.
- BER bit error rate
- the outer code consists of k repetition codes with variable rates. As shown in FIG. 3 , suppose the rate of the ith repetition code is 1/d v (i) , the degree of the ith variable node of information bit is d v (i) . For the outer repetition codes, the EXIT curve is obtained in a similar manner as for RA codes and LDPC codes. Specifically, consider first un-punctured RZH codes. As discussed in Section 3, an information bit variable node of degree d v has d v incoming messages from the edge interleaver, and the decoder outputs are given by
- L rep out ⁇ ( i ) L ch + ⁇ j ⁇ i ⁇ L rep in ⁇ ( j ) , ( 3 )
- L ch is the LLR value about the information bit from the channel.
- L ch 2 ⁇ ⁇ x u ⁇ 2
- x u is the channel observation of the binary phase shift keying (BPSK) modulated information bit u.
- L rep in (j) is modeled as the output LLR of an AWGN channel whose input is the jth interleaver bit transmitted using BPSK.
- J ⁇ ( ⁇ ) ⁇ • ⁇ 1 - 1 2 ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ 2 ⁇ ⁇ - ⁇ ⁇ ⁇ e - ( ⁇ - ⁇ 2 / 2 ) 2 2 ⁇ ⁇ ⁇ 2 ⁇ log 2 ⁇ [ 1 + e - ⁇ ] ⁇ d ⁇ . Since J( ⁇ ) is a monotonically increasing function of ⁇ , it is invertible.
- the degree profile of a code can be specified by a polynomial
- an information bit variable node of degree d v has d v +1 incoming messages: d v from the edge interleaver and one from the channel.
- the decoder outputs are given by (3).
- the EXIT function of a degree-d v variable node is given by
- I E , VND ⁇ ( I A , VND , d v , E s N 0 ) J ⁇ ( ( d v - 1 ) ⁇ [ J - 1 ⁇ ( I A , VND ) ] 2 + ⁇ ch 2 ) ( 11 )
- I E , VND g ⁇ ( I A , VND , E s N 0 , R c , r ) ( 13 )
- EXIT charts can be used to analyze the convergence properties of RA codes and LDPC codes. The same principle can be applied to RZH codes. Specifically, if there is an open tunnel between the EXIT functions of the inner code and outer code for given r and
- the iterative decoding algorithm can asymptotically converge and then
- the EXIT functions for Hadamard codes with multiple inputs on BIAWGN channels are considered. Since it is difficult to directly compute the EXIT functions of the output from the MAP decoder, use is made of a pMAP decoder to derive an approximation for the extrinsic mutual information over binary-input additive white Gaussian noise BIAWGN channels. Basic properties of BIAWGN channel are introduced and the pMAP decoder, with which an approximate expression for the EXIT function over BIAWGN for general linear block codes and Hadamard codes is derived.
- a BIAWGN channel is characterized by its variance ⁇ 2 .
- I i,1 ⁇ ⁇ j
- c j,i 0 ⁇ the set of indices of all codewords in the dual code with the i-th bit being 0.
- the extrinsic MAP decoding can be implemented by using the dual code as follows
- c _ 0 ⁇ ⁇ log 2 ⁇ ( 1 + T EMAP , i ) ⁇ E T EMAP , i
- c _ 0 ⁇ ⁇ log 2 ⁇ ( D EMAP , i ) ⁇ . Since D EMAP D EpMAP , we further have
- I E i BIAWGN E T EMAP , i
- c _ 0 ⁇ ⁇ log 2 ⁇ ( D EpMAP , i ) ⁇ E T _
- c _ 0 ⁇ ⁇ log 2 ⁇ ( ⁇ S ⁇ ⁇ ⁇ ⁇ ⁇ P ⁇ ( I i , 1 ⁇ ) ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ( 1 + ⁇ l 1 , ⁇ l ⁇ i ⁇ n ⁇ T l ⁇ j ⁇ S ( ⁇ c j , l ⁇ ) ) ( - 1 ) S
- + 1 ) ⁇ E T _
- c _ 0 ⁇ ⁇ ⁇ S ⁇ P ⁇ ( I i , 1 ⁇ ) ⁇ ⁇ ⁇ ⁇ ( - 1 ) ⁇ S ⁇ + 1 ⁇ log 2 ⁇ ( 1 + ⁇ l 1 , ⁇ l ⁇ i ⁇ n
- V(S) ⁇ l ⁇ 1, . . .
- Theorem 4.2 can be used to approximately compute the extrinsic information at the output of the MAP decoder for a linear block code.
- Hadamard codes are low rate codes.
- the extrinsic information can be approximated by I E,i BIAWGN ( I A,1 , . . . , I A,n ) ⁇ 1 ⁇ I E,i ⁇ ,BIAWGN (1 ⁇ I A,1 , . . . , 1 ⁇ I A,n ), (30) where I E,i ⁇ ,BIAWGN (1 ⁇ I A,1 , . .
- c j [c j,1 , c j,2 , . . . , c j,n ] T denote the j-th codeword of the Hadamard code over GF(2).
- I i,1 H,d ⁇ j
- G H,r is in systematic form
- extrinsic and communication channels are AWGN channels with respective mutual information I A,d , I A,p , I A,q and I A,p m , we have
- extrinsic and communication channels are AWGN channels with respective mutual information I A,d , I A,p and I A,p m , then we have
- extrinsic and communication channels are AWGN channels with respective mutual information I A,q , I A,d and I A,p .
- the outer code consists of k repetition codes with variable rates and the EXIT functions of repetition decoders are identical to the case where the serial decoding is employed. Also, since the connection bit nodes are viewed as variable nodes with degree 2, the corresponding EXIT functions for q and p m are simply the EXIT functions for rate 1 ⁇ 2 repetition code without message from the channel. Now we have both the EXIT functions of the outer repetition decoder and the inner Hadamard decoder, we can then trace the mutual information exchanged between the two constituent components for a given degree profile and calculate the corresponding BER after a given number of iterations as in [?], with which we are able to perform degree profile optimization by using differential evolution (DE).
- DE differential evolution
- I ch be the mutual information about the LLRs of the coded bits transmitted from the communication channel.
- I A,a H , I A,q H , I A,p m H be the mutual information about the a priori LLRs passing from the repetition decoder to the Hadamard decoder (the extrinsic channels) for the information bits, connection bit and the last parity bit, respectively, which are initialized to zero for the first iteration.
- the output mutual information of the Hadamard decoder will then serve as the input I A,VND for the EXIT functions of the corresponding repetition decoder, which will produce the corresponding I E,VND and feed back to the EXIT functions of the Hadamard decoder as the new inputs for the next iteration.
- the mutual information passed to the outer repetition decoder is I A,VND , then the BER is given by
- the optimal IRZH ensemble parameters ⁇ a i ⁇ are the one that minimize E b /N 0 , subject to a vanishing BER P b .
- the vanishing BER condition is often replaced by a BER threshold ⁇ for a given iteration number N it , i.e., P b (N it ) ⁇ , which leads to the following optimization problem
- the essential idea behind DE is a self-organizing scheme to generate a trial parameter vector (which is ⁇ a i ⁇ in our case) by adding the weighted difference between two population vectors of the current generation to a third vector which is the target vector. If the trial vector has a smaller cost value (which is P b in our case and can be estimated by using the EXIT functions of the component codes as discussed in Section 1) than the target vector, it survives into the next generation of the evolution. After a given number of evolutions, the vector with the smallest cost value among the population of current generation is the optimized parameter vector. Note that although there is no mathematical convergence proof of DE, it is believed to be a simple and reliable global optimization method.
- E b N 0 ⁇ 0.98 dB is also plotted 700 in FIG. 7 , where the solid line denotes the EXIT function of the outer mixture repetition codes with the optimized profile and diamond denotes the EXIT function of the inner ZH code.
- the EXIT chart of the curve-fitting result 900 is shown in FIG. 9 and the BER performance is shown 1001 in FIG. 10 .
- Results show that the simulated SNR threshold is ⁇ 1.14 dB (only around 0.31 dB away from the capacity) which is much better than that of the optimized un-punctured code with the same r and a much lower (0.0179) coding rate.
- the method for code design for IRZH codes is detailed by flow chart 40 A in FIG. 4A for serial decoding and flow chart 50 A in FIG. 5A for parallel decoding.
- the method is a mutual-information-based code design for both the serial and parallel decoding schemes.
- the decoding of RZH code is based on its tanner graph 30 which is depicted in FIG. 3 .
- the inner decoder utilizes the trellis structure of the zigzag-Hadamard code and the common bits participate in the two-way decoding of ZH codes. Extrinsic information about the information bits is exchanged between the inner ZH decoder and outer repetition decoder, see FIG. 4 .
- Chart 40 A details the steps for IRZH code design for serial decoding.
- Step 1 a small signal to noise ratio Es/N 0 is initially chosen 42 A.
- Step 2 for a given Hadamard order r and Es/N 0 , the EXIT curve for ZH code is obtained by Monte Carlo simulation using serial decoding 43 A.
- Step 3 with the obtained EXIT curve there is an attempt to find the optimal IRZH ensemble parameters ⁇ a i ⁇ that maximize coding rate subject to vanishing BER by solving the following optimization problem
- the vanishing BER condition can be implemented by checking the existence of a decoding tunnel between the EXIT curve of the ZH code obtained in step 2 and that of outer repetition codes. Namely, the curve defined by a particular degree profile lies below the EXIT curve of ZH code in the EXIT chart.
- Step 4 if the maximum rate obtained by step 3 is smaller than the target rate, the signal to noise ratio Es/N 0 is increased by a small amount and step 2 and 3 are repeated, until the obtained achievable rate equals to the target rate 45 A.
- variable nodes of common bits are viewed as degree-2 information bit nodes and extrinsic LLRs about information bits and common bits are parallel exchanged between the outer repetition decoder and inner Hadamard decoder.
- Chart 50 A details the steps for IRZH code design for parallel decoding.
- Step 1 a relatively large signal to noise ratio SNR E b /N 0 is initially chosen 52 A.
- Step 2 for a given Hadamard order r coding rate R c and SNR, a degree profile that has minimum BER by using deferential evolution (DE) is found 53 A.
- DE deferential evolution
- BER is computed, which can be done by a mutual information tracking scheme.
- EXIT functions of the Hadamard codes in AWGN channel are derived, with which the BER can be easily estimated 54 A. Specifically, the EXIT functions for the information bits are given by:
- Step 4 if the minimum BER obtained by step 2 is smaller than the target BER requirement, the signal to noise ratio SNR E b /N 0 is reduced and step 2 is repeated, until the target BER requirement cannot be achieved 55 A.
- the least SNR that can still achieve the BER requirement is the minimum SNR and the corresponding degree profile is the optimized degree profile.
- the present code design method is optimized for IRZH codes with serial and parallel decoding schemes.
- serial decoding the EXIT functions are obtained via simulation; on the other hand, for parallel decoding, the EXIT functions of Hadamard codes in the context of RZH codes are derived.
- BERs for a given code profile can be easily estimated and either curve fitting (for serial decoding) or differential evolution (DE) technique (for parallel decoding) can be employed to optimize the degree profile for the given design parameters, e.g., the coding rate and the Hadamard order.
- DE differential evolution
- the techniques introduced in this report can also be easily extended to the design of general low-density parity-check codes where the check nodes are replaced by other linear block codes, e.g., Hadamard codes or Hamming codes.
- the present code design method for irregular repeat-zigzag-Hadamard (IRZH) codes to approach capacity with turbo-like serial decoding or LDPC-like parallel decoding is quite different from current techniques and provides significant advantages.
- a check node of a LDPC code is a single parity-check constraint.
- the design procedure for LDPC codes is similar to that of IRZH codes with parallel decoding. The difference is that with the present method it is a common practice to use the analytical EXIT function of the inner check node over BEC to approximate that over BIAWGN channel. Results show that such an approximation works very well for irregular LDPC code design.
- the present EXIT-function-based design method offers high quality design results whereas the optimization complexity is low. Note that although density-evolution can be used for the design of IRZH code, the complexity is very high.
- the inner decoder is a trellis-based zigzag-Hadamard (ZH) decoder
- ZH zigzag-Hadamard
- connection bits of the ZH code are treated as degree-2 variable nodes, and the inner constituent decoder becomes a Hadamard decoder.
- simulation can still be utilized to obtain the EXIT function
- the present method derives analytical EXIT functions of Hadamard codes with multiple inputs and outputs in the context of low-rate RZH codes over BIAWGN channels, and applies these results to the analysis and design of RZH codes. With analytical functions, the optimization procedure is further simplified.
- Test results show that the present design method works well and serves as practical tool for the low-rate IRZH code optimization.
- the present design method can also be extended to general LDPC codes where the single parity-check nodes are replaced by other linear codes, e.g., Hadamard codes or Hamming codes.
Abstract
Description
where r is the order of the Hadamard code for the inner encoder.
With k→∞, then
When the coding rate is very low, e.g., Rc≦0.01, the performance difference between the punctured one and the un-punctured code is relatively small; whereas in the range around Rc=0.05, the punctured code offers a significant coding gain over the un-punctured one due to the elimination of the repetition bits embedded in the inner ZH codes.
where Lch is the LLR value about the information bit from the channel. For AWGN channel with noise variance
then
where xu is the channel observation of the binary phase shift keying (BPSK) modulated information bit u. Lrep in(j) is modeled as the output LLR of an AWGN channel whose input is the jth interleaver bit transmitted using BPSK. The EXIT function of a degree-dv variable node is then
I E,VND(I A,VND ,d v)=J(√{square root over ((d v−1))}J −1(I A,VND)) (4)
where
Since J(σ) is a monotonically increasing function of σ, it is invertible.
where ai≧0 is the fraction of nodes having degree dv,i. Note that {ai} must satisfy
The average variable node degree is then given by
Since for
then
Let bi be the fraction of edges incident to variable nodes of degree dv,i. There are (kai)dv,i edges involved with such nodes, so then
Note that {bi} must satisfy
It has been shown that the EXIT function of a mixture of codes is an average of the component EXIT functions. With (4), (6) and (7), the effective transfer function of the outer mixture codes is thus given by
Since g(·) is a monotonically increasing function of IA,VND, it is invertible and then IA,VND=g−1(IE,VND,Rc,r).
where
being the symbol SNR. For systematic RZH codes, then
hence
Then with (7), (11) and (12), one can have the effective transfer function of the outer mixture codes as follows
Again, g(·) is invertible, and one can have
In order to perform code ensemble optimization, the EXIT transfer function of the inner ZH decoder is needed. In general, there is no closed-form formula, hence the EXIT function is computed by simulation and denoted it by
Code Ensemble Optimization
the optimal IRZH ensemble parameters {ai} that maximize Rc subject to vanishing BER
are solution of the following optimization problem
EXIT charts can be used to analyze the convergence properties of RA codes and LDPC codes. The same principle can be applied to RZH codes. Specifically, if there is an open tunnel between the EXIT functions of the inner code and outer code for given r and
the iterative decoding algorithm can asymptotically converge and then
with a sufficient large number of iteration. Mathematically, the convergence condition is given by
for un-punctured RZH codes, and
for systematic codes. With the help of the EXIT chart analysis, one can simplify the above optimization problem by replacing the constraint
in (17) with condition (18) or (19). To design an IRZH code for a particular rate Rc, we simply choose proper Hadamard degree and change
in (17) until the resulting optimization result Rc*=Rc.
Design Method for LDPC-Like Decoding
denote the LLR and “soft bit” estimation of X based on the channel observation Y respectively. For a BIAWGN channel one can have
Hence (T|X=+1) has the following pdf
where fL denotes the pdf of (L|X=+1). With (20), the (2i)-th moment of (T|X=+1) as a function of m is given by
The mutual information between X and L as a function of m is given by [?]
which is invertible and hence has m=W−1(I).
MAP and pMAP Decoders
Let DEMAP,i be the extrinsic output of MAP decoder (23) after the following modification
The extrinsic output of a suboptimal decoder which one can call pMAP decoder [?] is defined as
where P(Ii,1 ⊥) denotes the set of all subsets of Ii,1 ⊥ and “” denotes logical “or” operation. In what follows, the pMAP decoder is employed to obtain an approximate expression for the EXIT functions of linear block codes over a BIAWGN channel. Since the basic idea is to decompose the EXIT function in the form of a series EXIT functions of multiple single parity-check codes, we will start from the EXIT function of single parity-check codes.
EXIT Functions for General Linear Block Codes
then using a known proposition 2.6, the extrinsic mutual information of the i-th bit at the output of MAP decoder is given by
where the last equality holds since the channel is assumed to be memoryless.
For the single parity-check codes, pMAP and MAP decoders are identical in any memoryless channel, and the Conjecture 4.1 also states that for other high rate codes, DEMAP≅DEpMAP when the all-zero codeword is transmitted over the BIAWGN channel. We have the following result.
where V(S)={lε{1, . . . , n}| jεScj,l ⊥−1}, and ml=W−1(IA,l) where IA,l is the a priori mutual information about Xl from the channel given Xl=+1 is transmitted.
Since DEMAP=DEpMAP, we further have
Define V(S)={lε{1, . . . , n}| jεScj,l ⊥=1}, then the E{·} term in (29) is the extrinsic information of a (|V(S)|,|V(S)|−1) single parity-check code. Since all-zero codeword is transmitted, according to Theorem 4.5, we have
with which (29) becomes
where ml is the mean of the “soft bit” estimation passed from the channel to the MAP decoder for Xl given that X=+1 is transmitted. Note that such a BIAWGN channel can either be a communication channel or an extrinsic channel. It is also noted that the EXIT functions in the form of multiple EXIT functions of the same code in BEC channel, cannot be expanded.
I E,i BIAWGN(I A,1 , . . . , I A,n)≅1−I E,i ⊥,BIAWGN(1−I A,1, . . . , 1−I A,n), (30)
where IE,i ⊥,BIAWGN(1−IA,1, . . . , 1−IA,n) is the extrinsic information of its high-rate dual code and can be computed by Theorem 4.2 with ml=W−1(1−IA,l).
EXIT Functions for Hadamard Codes over BIAWGN Channel
Where IE,d
Then we have the following results for Hadamard codes that will be subsequently used:
for all possible {dl}l≠i. Since G H,r is in systematic form, g i can be written as [u i,p i] where u i=[u0, u1, . . . , ur] with ul=0 for l≠i and ui=1. By switching the rows g i and g l, we obtain G H,r (1) with g (1)=g l and g l (1)=g i. It is clear that the subcode Ci (1) generated by G H,r (1) is identical to Ci. We further construct G H,r (2) from G H,r (1) by letting (g i (2)=g i (1)+(g l (1)+g i (1))=g l (1)=g i, and g l (2)=g l (1)+(g l (1)+g i (1))=g i (1)=g l, by which we have G H,r (2)=G H,r (1) and hence Ci (2)=Ci. The effect of transforming G H,r (1) to G H,r (2) is only to change the order of the coded bits. Since both codes are systematic codes, there is one order change between systematic bit position i and l (note that there are only two l's in u i+u l), and all the other changes are happened between parity bit positions. It is clear that Ci (2)=Ci is the ·Cl* we are looking for and this concludes the proof.
where mq=W−1(1−IA,q), md=W−1(1−IA,d), mp=W−1(1−IA,p) and Mp
Then similar to Theorem 4.3, the mutual information of q can be computed by the following result.
where md=W−1(1−IA,d), mp=W−1(1−IA,p) and mp
by which we can compute the EXIT function for pm as follows.
where mq=W−1(1−IA,q), md=W−1(1−IA,d) and mp=W−1(1−IA,p).
Also let IA,a H, IA,q H, IA,p
where
Code Ensemble Optimization
To solve (17) for irregular RZH codes with parallel decoding, we resort to the differential evolution (DE) algorithm, which is a powerful population-based genetic algorithm and was introduced for finding the optimal degree profile of the irregular LDPC codes. The essential idea behind DE is a self-organizing scheme to generate a trial parameter vector (which is {ai} in our case) by adding the weighted difference between two population vectors of the current generation to a third vector which is the target vector. If the trial vector has a smaller cost value (which is Pb in our case and can be estimated by using the EXIT functions of the component codes as discussed in Section 1) than the target vector, it survives into the next generation of the evolution. After a given number of evolutions, the vector with the smallest cost value among the population of current generation is the optimized parameter vector. Note that although there is no mathematical convergence proof of DE, it is believed to be a simple and reliable global optimization method. To find the solution to (17) for given Rc and r, we first choose an Eb/N0 and find a degree profile {ai} using DE that satisfies the BER constraint ε. If at least one {ai} exists, the Eb/N0 is reduced and the procedure is repeated until no such a vector {ai} exists. The smallest value of Eb/N0 for which a power profile {ai} satisfies the BER constraint is the desired minimum SNR and the corresponding {ai} is the optimized degree profile.
Numerical Results
where the degree profile polynomial is given by f(x)=0.323x3+0.175x8+0.026x22+0.476x23. The corresponding EXIT chart for
−0.98 dB is also plotted 700 in
with f(x)=0.363x3+0.338x6+0.299x7, and the simulated threshold (measured at Pb=10−4) is −1.13 dB, which is only 0.41 dB away from the capacity.
with f(x)=0.467x3+0.375x10+0.158x11, which is only 0.1 dB away from the capacity. The EXIT chart of the curve-fitting result 900 is shown in
with f(x)=0.364x3+0.014x11+0.192x12+0.074x13+0.095x36+0.136x37+0.1087x38+0.004x39+0.013x200, and the simulated SNR threshold is around −1.2 dB, which is 0.34 dB away from the capacity and around 0.8 dB better than the same rate un-punctured code with r=6. With r=6, a code is optimized at
with f(x)=0.401x3+0.573x7+0.026x20, and the simulated SNR threshold is −1.24 dB, which is slightly better than that of r=4.
This problem can be solved by any linear optimization program 44A. The vanishing BER condition can be implemented by checking the existence of a decoding tunnel between the EXIT curve of the ZH code obtained in
Design for Parallel Decoding
similarly, the EXIT function for the connection bits can be computed by
and the EXIT function for the last parity bit is given by
where the details of the variables in the above equations are explained elsewhere herein. Step 4: if the minimum BER obtained by
Claims (17)
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